A policyholder has probability 0.7 of having no claims, 0.2 of having exactly one claim, and 0.1 of having exactly two claims. Claim amounts are uniformly distributed on the interval [0, 60] and are independent. The insurer covers 100% of each claim.

Calculate the probability that the total benefit paid to the policyholder is 48 or less.

• 0.320
• 0.400
• 0.800
• 0.892
• 0.924

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: D

Consider the three cases based on the number of claims. If there are no claims, the probability of total benefits being 48 or less is 1. If there is one claim, the probability is 48/60 = 0.8, from the uniform distribution. If there are two claims, the density is uniform on a 60x60 square. The event where the total is 48 or less is represented by a triangle with base and height equal to 48. The triangle’s area is 48x48/2 = 1152. Dividing by the area of the square, the probability is 1152/3600 = 0.32. Using the law of total probability, the answer is 0.7(1) + 0.2(0.8) + 0.1(0.32) = 0.892.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.